Regression Lines from Computer Output and R^2

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Reading Computer

Output and r²

Chapter 3.2

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Reading Computer Output

Many times in this course, you will not be asked to create the least-squares regression line using your calculator, especially on the AP Exam. This process is time-consuming and you can easily make a small mistake typing in the data that will affect your answer. We want to know if you understand the results, not if you can punch in numbers.

In the real world, statisticians use computers and specialized computer software to analyze data, not graphing calculators. The next slide shows an example of output that might be generated from one of these programs.

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Reading Computer Output

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Example

Desiree is interested to see if students who consume more caffeine tend to study more as well. She randomly selects 20 students at her school and records their caffeine intake (mg) and the number of hours spend studying. A scatterplot of the data showed a linear relationship. The computer output from a least-squares regression analysis of the data is below:

A.

Write the prediction equation in the context of the problem.

Multiple Choice

Desiree is interested to see if students who consume more caffeine tend to study more as well. She randomly selects 20 students at her school and records their caffeine intake (mg) and the number of hours spend studying. A scatterplot of the data showed a linear relationship. The computer output from a least-squares regression analysis of the data is shown.

Write the regression equation.

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Example

Desiree is interested to see if students who consume more caffeine tend to study more as well. She randomly selects 20 students at her school and records their caffeine intake (mg) and the number of hours spend studying. A scatterplot of the data showed a linear relationship. The computer output from a least-squares regression analysis of the data is below:

B.

Interpret the slope in the context of the problem.

Open Ended

Desiree is interested to see if students who consume more caffeine tend to study more as well. She randomly selects 20 students at her school and records their caffeine intake (mg) and the number of hours spend studying. A scatterplot of the data showed a liner relationship. The computer output from a least-squares regression analysis of the data is shown.

Interpret the slope in the context of the problem.

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Example

Desiree is interested to see if students who consume more caffeine tend to study more as well. She randomly selects 20 students at her school and records their caffeine intake (mg) and the number of hours spend studying. A scatterplot of the data showed a linear relationship. The computer output from a least-squares regression analysis of the data is below:

C.

How many hours of studying would you predict for a person who consumes 200 mg of caffeine?

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Example

Desiree is interested to see if students who consume more caffeine tend to study more as well. She randomly selects 20 students at her school and records their caffeine intake (mg) and the number of hours spend studying. A scatterplot of the data showed a linear relationship. The computer output from a least-squares regression analysis of the data is below:

D

Andre drank 2 cans of Celcius (approx 400 mg of caffeine) and studied for 8 hours. What is his residual?

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R-Squared

Notice that the computer output gives a value for r² and not a value for r. r² is called the coefficient of determination . It tells us the proportion of variation in the response (y) variable that can be explained by its relationship with the predictor (x) variable. Think about it this way:

Data was collected about study time before a test and the resulting test grade.  The resulting regression equation was:  predicted score = 42 + 2.5(study time), r² = 0.74


Do you think that the amount of time a student studies is the only thing that can explain how well they do on a test? Probably not. The r² value says that if there is a linear relationship, then about 74% of the variation in the test scores can be explained by study time, and the other 26% would be explained by something else.

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How do we find correlation from computer data?

To find the
correlation
coefficient (r), from
computer output,
take the square
root of r². If the
slope is positive, r
should be positive.
If the slope is
negative, r should
be negative.

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Example

Desiree is interested to see if students who consume more caffeine tend to study more as well. She randomly selects 20 students at her school and records their caffeine intake (mg) and the number of hours spend studying. A scatterplot of the data showed a linear relationship. The computer output from a least-squares regression analysis of the data is below:

E.

What is the correlation coefficient? Interpret this value.

Open Ended

Desiree is interested to see if students who consume more caffeine tend to study more as well. She randomly selects 20 students at her school and records their caffeine intake (mg) and the number of hours spend studying. A scatterplot of the data showed a linear relationship. The computer output from a least-squares regression analysis of the data is shown.

What is the correlation coefficient? Interpret this value.

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Example

Desiree is interested to see if students who consume more caffeine tend to study more as well. She randomly selects 20 students at her school and records their caffeine intake (mg) and the number of hours spend studying. A scatterplot of the data showed a linear relationship. The computer output from a least-squares regression analysis of the data is below:

F.

What proportion of observed variation in the response variable can be attributed to the approximate linear relationship between the variables? Interpret this value.

Open Ended

Desiree is interested to see if students who consume more caffeine tend to study more as well. She randomly selects 20 students at her school and records their caffeine intake (mg) and the number of hours spend studying. A scatterplot of the data showed a linear relationship. The computer output from a least-squares regression analysis of the data is shown.

What proportion of observed variation in the response variable can be attributed to the approximate linear relationship between the variables? Interpret this value.